Problem: Simplify the following expression: $r = \dfrac{-42n^3}{-60n^3 + 78n^2}$ You can assume $n \neq 0$.
Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-42n^3 = - (2\cdot3\cdot7 \cdot n \cdot n \cdot n)$ The denominator can be factored: $-60n^3 + 78n^2 = - (2\cdot2\cdot3\cdot5 \cdot n \cdot n \cdot n) + (2\cdot3\cdot13 \cdot n \cdot n)$ The greatest common factor of all the terms is $6n^2$ Factoring out $6n^2$ gives us: $r = \dfrac{(6n^2)(-7n)}{(6n^2)(-10n + 13)}$ Dividing both the numerator and denominator by $6n^2$ gives: $r = \dfrac{-7n}{-10n + 13}$